Purpose: To introduce students to related rates problems building on their understanding of the chain rule.
Classroom Procedure: Students should be given time to work on the multiple choice portion of the first question in small groups and then this should be discussed in detail with the whole class. This is the place where students are making the connection between the chain rule that they have learned in Module II and the fact that this gives us a relationship between different rates of change that can be helpful in answering other questions. After this discussion class time should alternate between students working on the remaining problems in small groups and whole class discussions on these problems when it seems appropriate given the misconceptions or struggles that are happening in those small groups. This activity will either require some significant work on the students part outside of class or two days of class time.
Ideas this Activity Builds On: This activity mainly builds on the students understanding of the chain rule that was developed in Module II. This activity also builds on the work we have already done on translating from narrative representation into symbolic representation. There is no need of implicit differentiation here.
Introduction/Motivation of the Activity: To use our understanding of the chain rule to help take advantage of our knowledge about one rate of change to find another rate of change.
Need to Establish by the End of Activity/Wrap-Up: How to identify a relationship between two changing quantities, write down that relationship symbolically and move from that relationship to a relationship between the rate at which those quantities are changing.
Additional Notes: There are no synthesis questions for this activity. In many ways the next activity serves as the synthesis for this activity as we move to related rates questions that require implicit differentiation. Implicit differentiation will be introduced within that activity.